For the Regge-Wheeler gravitational wave equation potential, \(V_S(x)\)  is specified with \(s=2\) . The contribution from \(V_S(x)\)  in the interior of a 2-2-hole is completely negligible, and so we display \(V_R(x)\)  in the interior in Fig. REF . This figure shows that \(V_R(x)\)  has a maximum value of \(\approx 0.014/M^2\) .\(V_R(x)\)  for a scalar wave equation has the oppositie sign and it was displayed in [1]}. This can be compared to the maximum value of \(V_S(x)\)  that occurs at the minimum value of \(S(x)\) , that is at \(x=x_{\rm lr}\) . This potential barrier height is \(l(l+1)/(27M^2)\)  where \(l\ge s=2\) . \(V_R(x)\)  is negative in this region and has a minimum value of \(-81/(1024M^2)\)  at \(x=x_{\rm lr}-(2\log (3/2)+1/3)M\) , but \(V_S(x)\)  dominates. Thus the potential generated in the interior of the 2-2-hole is smaller, but not negligible, compared to the standard black hole potential barrier around the light ring. This small interior potential is the only impact that the large curvatures have on the wave equation. In addition the wavelengths of interest are often comparable or larger than the radial extent of the interior. The gravitational wave can also interact with the matter in the interior as discussed in [2]}.
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